Editing Isoperimetric dimension (section)

==Examples==

A ''d''-dimensional Euclidean space has isoperimetric dimension ''d''. This is the well known [[isoperimetry|isoperimetric problem]] &mdash; as discussed above, for the Euclidean space the constant ''C'' is known precisely since the minimum is achieved for the ball. 

An infinite cylinder (i.e. a [[cartesian product|product]] of the [[unit circle|circle]] and the [[real line|line]]) has topological dimension 2 but isoperimetric dimension 1. Indeed, multiplying any manifold with a compact manifold does not change the isoperimetric dimension (it only changes the value of the constant ''C''). Any compact manifold has isoperimetric dimension&nbsp;0.

It is also possible for the isoperimetric dimension to be larger than the topological dimension. The simplest example is the infinite [[jungle gym]], which has topological dimension 2 and isoperimetric dimension 3. See [https://web.archive.org/web/20040817075143/http://www.math.ucla.edu/~bon/jungle.html] for pictures and Mathematica code.

The [[hyperbolic geometry|hyperbolic plane]] has topological dimension 2 and isoperimetric dimension infinity. In fact the hyperbolic plane has positive [[Cheeger constant]]. This means that it satisfies the inequality

:<math>\operatorname{area}(\partial D)\geq C\operatorname{vol}(D),</math>

which obviously implies infinite isoperimetric dimension.